Next: Filter Design
Up: Prucha and Biondi: STANFORD
Previous: Karpushin: REFERENCESRemoving velocity stack
To build a velocity stack, we usually construct an operator 229#229that maps
energy from velocity-stack space to offset-travel time
space. The adjoint operator
395#395 maps the energy back to a velocity-stack space. To build the model
in a 394#394
domain that is consistent with the data, we then solve the least-squares problem:
This problem can be solved using iterative methods. For a simple model
the
solution is usually obtained in a few iterations. Although
this
solution may fit the data well, it may have ``butterfly'' artifacts
caused by
a limited aperture of the data. To illustrate this, I create a simple
model with one spike in the velocity-stack space.
Applying the operator 229#229 to this model, I obtain synthetic data
set. Then I use this modeled data as an
input (9#9) to solve the least-squares problem (equation ) for 10#10.Figure shows the original model in the velocity-stack
space, the modeled data, the estimated model, and data residual after five
iterations of the conjugate gradient method.
spike
Figure 1 Top left: Ideal model. Top
right: Spike after 5 iterations of the conjugate gradient. Bottom left:
Data modeled from the ideal model. Botton right: Data residual after
5 iterations
As Figure shows, the data residual after five iterations is small but ``butterfly'' artifacts are
clearly present in the solution.
It is desirable to obtain a model that has all the energy
concentrated at the
location of the original spike and fits the data well. This problem
has been addressed before. For example, at SEP ()
and ()
proposed to minimize the L1 norm of the model to
create the spiky solution. () solved a similar
problem using a parabolic Radon trasform, where they solve the problem in a
frequency domain.
These techniques showed very good results in concentrating the energy
of the solution.
(They are especially valuable during the multiple attenuation step of the processing.)
But for some applications it may be useful to have an inexpensive way to remove artifacts from
the model even if the data residual becomes larger.
My first goal is to use the spatial predictability of the artifacts to
design the operator to remove them. If the technique can make events appear better in a
velocity stack panel and is easy to apply, it can be useful in picking
velocities and designing masks for a multiple removal. A similar
approach may be effective in other geophysical applications where the
artifacts have a similar nature; for example, artifacts caused by a
limited apperture in Kirchhoff migration.
Next: Filter Design
Up: Prucha and Biondi: STANFORD
Previous: Karpushin: REFERENCESRemoving velocity stack
Stanford Exploration Project
6/7/2002